After the building explosion, when they are busted down to patrolman, Riggs and Murtaugh argue over which wire Riggs cut, having forgotten which wire he cut. During the incident, Riggs swapped which wire he was going to cut, first saying blue, then switching in the middle to red. In the cropped pan-and-scan TV version, it is unclear which wire he eventually cuts. However, in the original widescreen theatrical version, it is clear that Riggs cuts the red wire.
When Riggs is running out on the Ice at the hockey game he is hit by one of the players. When he is hit the second time and falls to the ice, there's a shot of him still in the hockey box struggling to get on the ice. This shot has obviously been put in the wrong spot. It should have been shown just after Riggs spotted Travis in the crowd, because when Riggs sees Travis he is standing inside the box.
The motorbike Riggs has commandeered apparently knows about its own imminent demise. As it plummets to the ground, its siren changes pitch and dies out before it actually hits the ground.
Near the end of the armored truck chase, Riggs stops the armored truck suddenly while in heavy traffic, sending a villain through the front window. But when Riggs gets out of the truck a second later, there is virtually no traffic on the road.
Leo was in protective custody because he was going to testify against the South Africans. But seeing as how Riggs and Murtaugh killed all of them, there was no need for Leo to testify or go into witness protection.
Q
What are the differences between the Theatrical Version and the Director's Cut?
A
Three dialogue/plot scenes were extended for the Director's Cut adding up nearly three minutes to the runtime. A detailed comparison between both versions with pictures can be found here.